Construction of weaving and polycatenane motifs from periodic tilings of the plane

Doubly periodic weaves and polycatenanes embedded in the thickened Euclidean plane are three-dimensional complex entangled structures whose topological properties can be encoded in any generating cell of its infinite planar representation. Such a periodic cell, called motif, is a specific type of link diagram embedded on a torus consisting of essential simple closed curves for weaves, or null-homotopic for polycatenanes. In this paper, we introduce a methodology to construct such motifs using the concept of polygonal link transformations. This approach generalizes to the Euclidean plane existing methods to construct polyhedral links in the three-dimensional space. Then, we will state our main result which allows one to predict the type of motif that can be built from a given planar periodic tiling and a chosen polygonal link method.

[1]  K. Shimokawa,et al.  On the Classification of Polyhedral Links , 2022, Symmetry.

[2]  M. Kotani,et al.  Classification of doubly periodic untwisted (p,q)-weaves by their crossing number and matrices , 2022, Journal of Knot Theory and Its Ramifications.

[3]  S. Hyde,et al.  Symmetric tangled Platonic polyhedra , 2022, Proceedings of the National Academy of Sciences.

[4]  Y. Tezuka,et al.  Topology of Polymers , 2019, SpringerBriefs in the Mathematics of Materials.

[5]  Akio Kawauchi,et al.  Complexities of a knitting pattern , 2018, Reactive and Functional Polymers.

[6]  S. Hyde,et al.  A Theoretical Schema for Building Weavings of Nets via Colored Tilings of Two-Dimensional Spaces and Some Simple Polyhedral, Planar and Three-Periodic Examples , 2018, Israel Journal of Chemistry.

[7]  O. Yaghi,et al.  The geometry of periodic knots, polycatenanes and weaving from a chemical perspective: a library for reticular chemistry. , 2018, Chemical Society reviews.

[8]  C. Adams,et al.  Hyperbolicity of links in thickened surfaces , 2018, Topology and its Applications.

[9]  C. Adams,et al.  Generalized bipyramids and hyperbolic volumes of alternating k-uniform tiling links , 2017, Topology and its Applications.

[10]  Banglin Chen,et al.  Some equivalent two-dimensional weavings at the molecular scale in 2D and 3D metal–organic frameworks , 2016 .

[11]  S. Hyde,et al.  Periodic entanglement III: tangled degree-3 finite and layer net intergrowths from rare forests. , 2015, Acta crystallographica. Section A, Foundations and advances.

[12]  J. Purcell,et al.  Volume bounds for weaving knots , 2015, 1506.04139.

[13]  Abhijit Champanerkar,et al.  Geometrically and diagrammatically maximal knots , 2014, J. Lond. Math. Soc..

[14]  Valeria Giardino,et al.  Forms and Roles of Diagrams in Knot Theory , 2014 .

[15]  Stephen T. Hyde,et al.  Periodic entanglement II: Weavings from hyperbolic line patterns , 2013 .

[16]  V. Robins,et al.  Periodic entanglement I: networks from hyperbolic reticulations , 2013 .

[17]  Dan Lu,et al.  The architecture of Platonic polyhedral links , 2009 .

[18]  K. Millett,et al.  The linking number and the writhe of uniform random walks and polygons in confined spaces , 2009, 0907.3805.

[19]  T. Endo,et al.  A graph-theoretic approach to a partial order of knots and links , 2008, 0806.3595.

[20]  S. Grishanov,et al.  DOUBLY PERIODIC TEXTILE STRUCTURES , 2008, 0806.2824.

[21]  Qiu Yuanyuan,et al.  Architecture of Platonic and Archimedean polyhedral links , 2008 .

[22]  S. A. Grishanov,et al.  KAUFFMAN-TYPE POLYNOMIAL INVARIANTS FOR DOUBLY PERIODIC STRUCTURES , 2007 .

[23]  D. Rolfsen.  Knots and Links , 2003 .

[24]  Ronald C. Read,et al.  The knot book: An elementary introduction to the mathematical theory of knots , 1997, Complex..

[25]  Osamu Kobayashi,et al.  Rotation numbers for curves on a torus , 1996 .

[26]  Benson Farb,et al.  A primer on mapping class groups , 2013 .

[27]  Guang Hu,et al.  Topological transformation of dual polyhedral links , 2010 .

[28]  Kumi Kobata,et al.  A circular embedding of a graph in Euclidean 3-space , 2010 .